zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Conjugacy invariants of Möbius transformations. (English) Zbl 0721.30035

This paper is devoted to generalizing the trace-squared (conjugation) invariant of Möbius transformations with complex entries to elements of a certain set of Clifford matrices 𝒞 n . These were first considered by K. Th. Vahlen [Math. Ann. 55, 585-593 (1902)] and later by L. V. Ahlfors [see e.g. Ann. Acad. Sci. Fenn., Ser. A I 105, 15-27 (1985; Zbl 0586.30045)]. The Clifford matrices considered here are slightly different from those obtained by Ahlfors and provide a generalization to orientation-reversing maps. The main theorem is that, for each k=0,1,2,···,n+2, the T k function (the definition is too complicated to give here)

T k :M(2,𝒞 n )

is a conjugation invariant. Here 𝒞 n is the algebra of Clifford numbers generated by n roots of -1. The conjugation is by Clifford matrices. Among the properties of the invariant T k are the following:

(i) T k (-g)=T(g) for all g𝒞 n . Thus T k is well-defined on the projectivization of 𝒞 n ·

(ii) T k (g)=0 for all k odd (resp. even) if g is orientation preserving (resp. reversing)

0 n+2 T k (g)=0·

(iii) g𝒞 n is hyperbolic if and only if T k (g)<0 for the largest k so that T k (g)0·

(iv) g𝒞 n is elliptic if and only if the quadratic form N(z):=zz ¯, when restricted to the kernel of Id-r(g) is not positive semidefinite. Here r takes g from the isometries of the ball model of hyperbolic space to the hyperboloid model.

MSC:
30G35Functions of hypercomplex variables and generalized variables